We describe a variety of new methods for accelerating construction of the Coulomb matrix that are based upon a simplified representation of the electron charge density. Central to these methods are the elimination of degeneracies, the reduction of complexity and a Hermite Gaussian representation. The utility of Hermite Gaussian-type functions as intermediates in the computation of the Coulomb matrix is stressed, and important consequences of this representation are pointed out, especially relationships to fast numerical methods. Projection methods are reviewed and new results obtained with two-center pontential-matched fits of the electron charge density are presented. Based on these results, and on simple arguments, we conclude that the use of single projection method to approximate the density is inefficient and introduces errors that cannot be controlled systematically. Hierarchical multipole methods and their quantum chemical adaptations are also reviewed. We introduce the fast Gauss transform to quantum chemistry and demonstrate its equivalence to a Cartesian multipole expansion in the asymptotic limit. A synthesis of the fast Gauss transform and a Barnes-Hut-like method is suggested and a first implementation involving only the Barnes-Hut method is described and shown to be highly competitive. In particular, calculations on large polypeptides and water clusters scale as favorably as N1.6, where N is the number of basis functions. An important feature of this new method is that accuracy is not sacrificed for speed, and that errors may be systematically controlled with a single thresholding parameter.